While many applications of automata in formal methods can use nondeterministic automata, some applications, most notably synthesis, need deterministic or good-for-games (GFG) automata.The latter Curtain are nondeterministic automata that can resolve their nondeterministic choices in a way that only depends on the past.The minimization problem for deterministic B"uchi and co-B"uchi word automata is NP-complete.In particular, no canonical minimal deterministic automaton exists, and a language may have different minimal deterministic automata.
We describe a polynomial minimization algorithm for GFG co-B"uchi word automata with transition-based acceptance.Thus, a run is accepting if it traverses a set $alpha$ of designated transitions only finitely often.Our algorithm is based on Mug a sequence of transformations we apply to the automaton, on top of which a minimal quotient automaton is defined.We use our minimization algorithm to show canonicity for transition-based GFG co-B"uchi word automata: all minimal automata have isomorphic safe components (namely components obtained by restricting the transitions to these not in $alpha$) and once we saturate the automata with $alpha$-transitions, we get full isomorphism.